Low-Dimensional Dynamics of Populations of Pulse-Coupled Oscillators

Pazó, Diego; Montbrió, Ernest

doi:10.1103/physrevx.4.011009

Cited by 149 publications

(200 citation statements)

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“…Only recently, spatiotemporally chaotic chimeras have been numerically identified in coupled oscillators on ring networks [45][46][47][48] and in globally connected populations of pulse-coupled oscillators [49]. However, a detailed characterization of the dynamical properties of these states have been reported only for the former case, more specifically, for rings of nonlocally coupled phase oscillators: in this case chimeras are transient, and weakly chaotic [46,50].…”

Section: Introductionmentioning

confidence: 99%

Chimera states in coupled Kuramoto oscillators with inertia

Olmi

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The dynamics of two symmetrically coupled populations of rotators is studied for different values of the inertia. The system is characterized by different types of solutions, which all coexist with the fully synchronized state. At small inertia the system is no more chaotic and one observes mainly quasiperiodic chimeras, while the usual (stationary) chimera state is not anymore observable. At large inertia one observes two different kind of chaotic solutions with broken symmetry: the intermittent chaotic chimera, characterized by a synchronized population and a population displaying a turbulent behaviour, and a second state where the two populations are both chaotic but whose dynamics adhere to two different macroscopic attractors. The intermittent chaotic chimeras are characterized by a finite life-time, whose duration increases as a power-law with the system size and the inertia value. Moreover, the chaotic population exhibits clear intermittent behavior, displaying a laminar phase where the two populations tend to synchronize, and a turbulent phase where the macroscopic motion of one population is definitely erratic. In the thermodynamic limit these states survive for infinite time and the laminar regimes tends to disappear, thus giving rise to stationary chaotic solutions with broken symmetry contrary to what observed for chaotic chimeras on a ring geometry.In 2002, simulations of abstract mathematical models revealed the existence of counterintuitive "chimera states", where an oscillator population splits into two parts, with one synchronizing and the other oscillating incoherently, even though the oscillators are identical. Since then, these counterintuitive states have become a relevant subject of investigation for experimental and theoretical scientists active in different fields, as testified by the rapidly increasing number of publications in recent years (for a review see [1, 2]). In this paper we analyze novel chimera states emerging in two symmetrically coupled populations of oscillators with inertia. In particular, the introduction of inertia allows the oscillators to synchronize via the adaptation of their own frequencies, in analogy with the mechanism observed in the firefly Pteroptix malaccae [3]. The modification of the classical Kuramoto model with the addition of an inertial term results in first order synchronization transitions and complex hysteretic phenomena [4][5][6][7][8][9]. Furthermore, networks of rotators have recently found applications in different technological contexts, including disordered arrays of Josephson junctions [10] and electrical power grids [11][12][13][14] and they could also be relevant for micro-electromechanical systems and optomechanical crystals, where chimeras and other partially disordered states likely play an important role with far reaching ramifications.

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Section: Introductionmentioning

confidence: 99%

Chimera states in coupled Kuramoto oscillators with inertia

Olmi

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The control scheme, like a tweezer, might be useful in experiments, where usually only small networks can be realized. Deeper analytical insight and bifurcation analysis of chimera states has been obtained in the framework of phase oscillator systems [42][43][44][45]. However, most theoretical results refer to the continuum limit only, which explains the behavior of very large ensembles of coupled oscillators.…”

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confidence: 99%

Tweezers for Chimeras in Small Networks

et al. 2016

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We propose a control scheme which can stabilize and fix the position of chimera states in small networks. Chimeras consist of coexisting domains of spatially coherent and incoherent dynamics in systems of nonlocally coupled identical oscillators. Chimera states are generally difficult to observe in small networks due to their short lifetime and erratic drifting of the spatial position of the incoherent domain. The control scheme, like a tweezer, might be useful in experiments, where usually only small networks can be realized. Deeper analytical insight and bifurcation analysis of chimera states has been obtained in the framework of phase oscillator systems [42][43][44][45]. However, most theoretical results refer to the continuum limit only, which explains the behavior of very large ensembles of coupled oscillators. In contrast, chimera states in small-size networks have attracted attention only recently [46][47][48][49], although in lab experiments usually only small networks can be realized.There are two principal difficulties preventing the observation of chimera states in small-size systems of nonlocally coupled oscillators. First, it is known that chimera states are usually chaotic transients which eventually collapse to the uniformly synchronized state [50]. Their mean lifetime decreases rapidly with decreasing system size such that one hardly observes chimeras already for 20 − 30 coupled oscillators. Moreover, a clear distinction between initial conditions that lead to a chimera state, and those that approach the synchronized state is no longer possible. Second, the position of the incoherent domain is not stationary but rather moves erratically along the oscillator array [51]. For large systems, this motion has the statistical properties of a Brownian mo- * corresponding author: schoell@physik.tu-berlin.de tion and its diffusion coefficient is inversely proportional to some power of the system size [51]. Both effects, finite lifetime and random walk of the chimera position, are negligible in large size systems. However, they dominate the dynamics of small-size systems, making the observation of chimera states very difficult. To overcome these difficulties some control techniques have been suggested recently. It has been shown that the chimera lifetime as well as its basin of attraction can be effectively controlled by a special type of proportional control relying on the measurement of the global order parameter [52]. On the other hand, Bick and Martens [53] showed that the chimera position can be stabilized by a feedback loop inducing a state-dependent asymmetry of the coupling topology. However, the latter control scheme relies on the evaluation of a finite difference derivative for some local mean field. This operation may become ill-posed for small system sizes like 20-30 oscillators, therefore one needs to use a refined control in this case.In this Letter, we propose an efficient control scheme which aims to stabilize chimera states in small networks. Like a tweezer, which helps to hold tiny objects...

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“…One proven modification is the inclusion of non-local effects of the geometry of the system that has been shown to lead to a co-existence of partially synchronized and partially asynchronized states of oscillators as a steady-state solution. Such states, addressed as chimera states, are the subject of recent theoretical and experimental studies (Kuramoto and Battogtokh, 2002;Abrams and Strogatz, 2004;Abrams et al, 2008;Ko and Ermentrout, 2008;Omel'chenko et al, 2008;Sethia et al, 2008;Sheeba et al, 2009;Laing, 2009a, b;Laing et al, 2012;Martens et al, 2013;Yao et al, 2013;Rothkegel and Lehnertz, 2014;Kapitaniak et al, 2014;Pazó and Montbrió, 2014;Panaggio and Abrams, 2014;Zhu et al, 2014;Gupta et al, 2014;Vasudevan and Cavers, 2014a, b). We focus our present study on defining a Kuramoto model with a phase lag that would accommodate the existence of chimera states.…”

Section: Mathematical Model Of the Earthquake Sequencingmentioning

confidence: 99%

Earthquake sequencing: chimera states with Kuramoto model dynamics on directed graphs

Vasudevan

Cavers

Ware

2015

Nonlin. Processes Geophys.

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Abstract. Earthquake sequencing studies allow us to investigate empirical relationships among spatio-temporal parameters describing the complexity of earthquake properties. We have recently studied the relevance of Markov chain models to draw information from global earthquake catalogues. In these studies, we considered directed graphs as graph theoretic representations of the Markov chain model and analyzed their properties. Here, we look at earthquake sequencing itself as a directed graph. In general, earthquakes are occurrences resulting from significant stress interactions among faults. As a result, stress-field fluctuations evolve continuously. We propose that they are akin to the dynamics of the collective behavior of weakly coupled non-linear oscillators. Since mapping of global stress-field fluctuations in real time at all scales is an impossible task, we consider an earthquake zone as a proxy for a collection of weakly coupled oscillators, the dynamics of which would be appropriate for the ubiquitous Kuramoto model. In the present work, we apply the Kuramoto model with phase lag to the nonlinear dynamics on a directed graph of a sequence of earthquakes. For directed graphs with certain properties, the Kuramoto model yields synchronization, and inclusion of nonlocal effects evokes the occurrence of chimera states or the co-existence of synchronous and asynchronous behavior of oscillators. In this paper, we show how we build the directed graphs derived from global seismicity data. Then, we present conditions under which chimera states could occur and, subsequently, point out the role of the Kuramoto model in understanding the evolution of synchronous and asynchronous regions. We surmise that one implication of the emergence of chimera states will lead to investigation of the present and other mathematical models in detail to generate global chimera-state maps similar to global seismicity maps for earthquake forecasting studies.

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Low-Dimensional Dynamics of Populations of Pulse-Coupled Oscillators

Cited by 149 publications

References 52 publications

Chimera states in coupled Kuramoto oscillators with inertia

Chimera states in coupled Kuramoto oscillators with inertia

Tweezers for Chimeras in Small Networks

Earthquake sequencing: chimera states with Kuramoto model dynamics on directed graphs

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